If $h(z)$ is analytic on the disk centered at 0 of radius r, by the Cauchy Residue formula
\[ \int \int_D h(z)\, dx dy = \pi r^2 h(0) \]
The disk is the simplest example of a quadrature domain since the integral of a holomorphic function over the domain is determined by the value at a single point.

How about the next simplest cases? What are connected quadrature domains whose integrals only depend on a few points (e.g. 2 or 3)?

\[ \int \int_D h(z)\, dx dy = c_1 h(z_1) + c_2 h(z_2) + c_3 h(z_3) \]

Probably these will all be close to the union of a few circles (with jumps in the coefficients as the radius changes).